Entrainment and chaos in driven nonlinear oscillators

Kevin Lin

Courant Institute.

Nonlinear oscillators are ubiquitous in physical, biological, and
engineered systems. Simple external forcing, for example periodic
pulse-like forcing, can dramatically modify the behavior of
oscillators and networks of oscillators, inducing a wide range of
responses which includes entrainment (phase-locking) and chaos.

The first part of this talk concerns a systematic computational
strategy for analyzing the dynamical behavior of pulse-driven
oscillators. This work builds on recent theoretical advances by Q.
Wang and L.-S. Young, who discovered and elucidated a general geometric
mechanism underlying these phenomena. Their theory predicts some
general dynamical features shared by a large class of pulse-driven
oscillators; the computational strategy proposed here provides
model-specific information and complements the theory. Throughout, I
will illustrate the main ideas via the Hodgkin-Huxley model, a
prototypical neuron model.

The second part of the talk concerns on-going work (joint with E.
Shea-Brown and L.-S. Young) on the response of small oscillator networks
to stochastic stimuli. This work is motivated by questions regarding
the ability of neural networks to respond reliably to repeated
presentations of complex signals. I will discuss preliminary
numerical results and what they tell us about geometric mechanisms
which can cause a network to behave unreliably.